Representations, duals and quantum doubles of monoidal categories

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[For the entire collection see Zbl 0742.00067.]The Tanaka-Krein type equivalence between Hopf algebras and functored monoidal categories provides the heuristic strategy of this paper. The author introduces the notion of a double cross product of monoidal categories as a generalization of double cross product of Hopf algebras, and explains some of the motivation from physics (the representation theory for double quantum groups).The Hopf algebra constructions are formulated in terms of monoidal categories C̲ and functors C̲→Vec̲ (finite-dimensional vector spaces) and generalized by replacing Vec̲ by another monoidal category V̲. It is interesting to remark that the monoidal category C̲ (functored over a category V̲) has a “Hopf algebra like structure”, V̲ having the role of the ground field. If V̲ is a quasitensor category, a coadjo!

@inProceedings{Majid1991, abstract = {[For the entire collection see Zbl 0742.00067.]The Tanaka-Krein type equivalence between Hopf algebras and functored monoidal categories provides the heuristic strategy of this paper. The author introduces the notion of a double cross product of monoidal categories as a generalization of double cross product of Hopf algebras, and explains some of the motivation from physics (the representation theory for double quantum groups).The Hopf algebra constructions are formulated in terms of monoidal categories $\underline\{C\}$ and functors $\underline\{C\}\rightarrow \underline\{\text\{Vec\}\}$ (finite-dimensional vector spaces) and generalized by replacing $\underline\{\text\{Vec\}\}$ by another monoidal category $\underline\{V\}$. It is interesting to remark that the monoidal category $\underline\{C\}$ (functored over a category $\underline\{V\}$) has a “Hopf algebra like structure”, $\underline\{V\}$ having the role of the ground field. If $\underline\{V\}$ is a quasitensor category, a coadjo!}, author = {Majid, Shahn}, booktitle = {Proceedings of the Winter School "Geometry and Physics"}, keywords = {Srni (Czechoslovakia); Proceedings; Winter school; Geometry; Physics}, location = {Palermo}, pages = {[197]-206}, publisher = {Circolo Matematico di Palermo}, title = {Representations, duals and quantum doubles of monoidal categories}, url = {http://eudml.org/doc/220868}, year = {1991},}

TY - CLSWKAU - Majid, ShahnTI - Representations, duals and quantum doubles of monoidal categoriesT2 - Proceedings of the Winter School "Geometry and Physics"PY - 1991CY - PalermoPB - Circolo Matematico di PalermoSP - [197]EP - 206AB - [For the entire collection see Zbl 0742.00067.]The Tanaka-Krein type equivalence between Hopf algebras and functored monoidal categories provides the heuristic strategy of this paper. The author introduces the notion of a double cross product of monoidal categories as a generalization of double cross product of Hopf algebras, and explains some of the motivation from physics (the representation theory for double quantum groups).The Hopf algebra constructions are formulated in terms of monoidal categories $\underline{C}$ and functors $\underline{C}\rightarrow \underline{\text{Vec}}$ (finite-dimensional vector spaces) and generalized by replacing $\underline{\text{Vec}}$ by another monoidal category $\underline{V}$. It is interesting to remark that the monoidal category $\underline{C}$ (functored over a category $\underline{V}$) has a “Hopf algebra like structure”, $\underline{V}$ having the role of the ground field. If $\underline{V}$ is a quasitensor category, a coadjo!KW - Srni (Czechoslovakia); Proceedings; Winter school; Geometry; PhysicsUR - http://eudml.org/doc/220868ER -